The CFKRS recipe prediction for symplectic shifted moments

Let $\mathcal{S}_{k}(X;S;M)$ be the twisted shifted moment defined in the source, where $S=\{s_1,\dots,s_k\}$, and for $J\subseteq\{1,\dots,k\}$ write $S_J=\{s_j:j\in J\}$ and $S_J^{-}=\{1-s_j:j\in J\}$. Define

$$T_M(S):=\frac{1}{2\zeta(2)}\sum_{Mn_1\cdots n_k=\square}\frac{a(Mn_1\cdots n_k)}{n_1^{s_1}\cdots n_k^{s_k}}.$$

CFKRS conjecture. If $|\operatorname{Re}(s)-\frac12|\ll1/\log X$ for all $s\in S$, then as $X\to\infty$,

$$\mathcal{S}_{k}(X;S;M)\sim\sum_{J\subseteq\{1,\dots,k\}}\prod_{j\in J}\mathcal{X}(s_j)X^{1+\frac{|J|}{2}-\sum_{j\in J}s_j}\widetilde g\left(1+\frac{|J|}{2}-\sum_{j\in J}s_j\right)T_M(S\setminus S_J\cup S_J^{-}).$$

This is the CFKRS prediction for quadratic Dirichlet-character moments of symplectic type; the paper presents it as an unproved asymptotic formula.